Theorem 0nnq 10847

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5665	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10835	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4022	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3917	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 197	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114  ∀wral 3051  ∅c0 4273   class class class wbr 5085   × cxp 5629  ‘cfv 6498  2^nd c2nd 7941  Ncnpi 10767   <_N clti 10770   ~_Q ceq 10774  Qcnq 10775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5148  df-xp 5637  df-nq 10835

This theorem is referenced by:  adderpq  10879  mulerpq  10880  addassnq  10881  mulassnq  10882  distrnq  10884  recmulnq  10887  recclnq  10889  ltanq  10894  ltmnq  10895  ltexnq  10898  nsmallnq  10900  ltbtwnnq  10901  ltrnq  10902  prlem934  10956  ltaddpr  10957  ltexprlem2  10960  ltexprlem3  10961  ltexprlem4  10962  ltexprlem6  10964  ltexprlem7  10965  prlem936  10970  reclem2pr  10971